论文标题
在$ \ mathbb {q} $上的椭圆曲线的显式打开图像
Explicit open images for elliptic curves over $\mathbb{Q}$
论文作者
论文摘要
对于在$ \ mathbb {q} $上定义的非CM椭圆曲线$ e $，Galois在其扭转点上的操作产生了Galois表示$ρ_e：GAL（\ edromline {\ Mathbb {Q}}}/\ Mathbb {Q}）达到同构。 Serre的著名定理说，$ρ_e$的图像是一个开放的，因此有限索引，$ gl_2的子组（\ wideHat {\ mathbb {z}}）$。我们描述了一种计算$ρ_e$的图像到$ gl_2（\ wideHat {\ mathbb {z}}}）$中的算法;该算法是实用的，并且已实施。最大程度地答案是对统一性问题的积极答案，并在有限数量的属属的明确模块化曲线上找到所有合理点，至少$ 2 $，我们将对这些组进行完整分类$ρ_e（gal（\ ovelline {\ Mathbb {q}}}}}}/\ MATHBB {Q}）\ CAP）\ CAP）\ CAP）\ CAP） sl_2（\ wideHat {\ mathbb {z}}）$和索引$ [gl_2（\ wideHat {\ mathbb {z}}）：ρ_e（ρ_e（gal（\ overline {\ Mathbb {Q}}}}}}}}}}}}/\ mathbb {q}） $ e/\ mathbb {q} $。本文的大部分内容都致力于通过以爱森斯坦序列表示的模块化形式对模块化曲线进行有效计算。
For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $ρ_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $ρ_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. We describe an algorithm that computes the image of $ρ_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least $2$, we give a complete classification of the groups $ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))\cap SL_2(\widehat{\mathbb{Z}})$ and the indices $[GL_2(\widehat{\mathbb{Z}}):ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))]$ for non-CM elliptic curves $E/\mathbb{Q}$. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.
